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The Sylow Subgroups of the Symmetric Group*

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The Sylow Subgroups of the Symmetric Group* THE SYLOWSUBGROUPS OF THE SYMMETRICGROUP* BY WILLIAM FINDLAY In the Sylow theorems f we learn that if the order of a group 2Í is divisible hj pa (p a prime integer) and not by jo*+1, then 31 contains one and only one set of conjugate subgroups of order pa, and any subgroup of 21 whose order is a power of p is a subgroup of some member of this set of conjugate subgroups of 2Í. These conjugate subgroups may be called the Sylow subgroups of 21. It will be our purpose to investigate the Sylow subgroups of the symmetric group of substitutions. By means of a preliminary lemma the discussion will be reduced to the case where the degree of the symmetric group is a power (pa) of the prime (p) under consideration. A set of generators of the group having been obtained, they are found to set forth, in the notation suggested by their origin, the complete imprimitivity of the group. The various groups of substitutions upon the systems of imprimi- tivity, induced by the substitutions of the original group, are seen to be them- selves Sylow subgroups of symmetric groups of degrees the various powers of p less than p". They are also the quotient groups under the initial group of an important series of invariant subgroups. In terms of the given notation convenient exhibitions are obtained of the commutator series of subgroups and also of all subgroups which may be consid- ered as the Sylow subgroups of symmetric groups of degree a power of p. Enumerations are made of the substitutions of periods p and p" and the con- jugacy relations of the latter set of substitutions are discussed. The subgroup consisting of all substitutions invariant under the main group is cyclic and of order p. The full set of conjugate Sylow subgroups of the symmetric group on p" letters fall into 7P-^- (a=j.«-1+J)«-J+---+J) + l) (piy classes each consisting of {(p — 2 ) ! }■ groups having the same complete system of imprimitivity. * Presented to the Society February 27, 1904. Received for publication June 20, 1903. tMathematische Annalen, vol. 5 (1873), page 584. 263 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 264 w. findlay: the sylow [July § 1. Lemma. We define the function it ( n, p ) of p any prime and n any integer, as the exponent of the highest power of p occurring as a factor of n !. If we impose the restriction that no power of p shall occur in the summation more than p — 1 times, n is uniquely expressible in the form n = '¿p"<, a^O. i=i We have the relation * n — k ■>r(n>P)=j^l- In particular Va — 1 "(pa>p) = j^ri> and therefore k Tr(n,p) = Y<Tr(pai,p)- § 2. Reduction of the problem. Throughout this paper, German capitals will be used to denote groups and the corresponding Boman capitals to represent substitutions; thus, 2i£ represents a group of degree a and order a and its substitutions will in general be denoted by Aj, A2, • • •. The symmetric group of degree n will be denoted by ©" and its Sylow subgroup of order a power of p, by ^jß". The letter E, with suffixes, will be used for the elements of ?ß which are considered as its generators. Our main problem is to find one subgroup ^5" of ©", having order p"^n'PK If a<c, any substitution, or group of substitutions on a letters — we will speak of the objects upon which the substitution is performed as letters — may properly be considered as an element or subgroup of &. Given a number of groups J*a, "0/3 ' ' ' "> í\c > if we consider the letters of the different groups as wholly distinct and the groups as subgroups of <Bm,m = a + b+.-- + k, then their least common overgroup consists of the totality of products of the form AB..-K and its order is a- ß •• -k. By the lemma of § 1, we can find a set of powers of p, say „ai „oj „a, P , P , , P , such that 1) « = ¿i>aS * Bachmann, Zahlentheorie, I, p. 33. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1904] SUBGROUPS OF THE SYMMETRIC GROUP 265 r 2) ■n-(n,p) = '£lTr(pai,p). i=X Distributing the n letters of <Sn into r sets containing p"x, p"1, • • •, p"r letters, respectively, we see that ty" is the least common overgroup of a set of Sylow subgroups * *P** (i = l,2,3,---,r). Since it is also a subgroup of !^pa, pa~>n, it may be considered sufficient to investigate only the case where n is a power of p. § 3. Generation of a $£,,., p). f We shall obtain ty1" by an inductive process, assuming the knowledge of a çqp«-x. Distribute the p" letters into p corresponding sets each containing p"~l letters. This we may indicate by the notation for the letters L4J (¿ = 1,2, ■■■,p;j = l, 2,3, •••,p-1). Let 21' be a ty1"'1 on the letters Ljtj. Transforming this by the p substitutions O _ /■^'l,l» ""1,2» ' ' " ' -kl.po-l \ (¿=1,2, ■■;p), ' \Li,l» Li,2, •*•» Lí,í>«-1/ we obtain a series of groups 21', 21", 21" ■• • $1^, whose least common overgroup D (§ 2) is of degree p" and order pP.*(.P"-x, p). Now ■>r(pa-l,p)=pa-2+?"-*+ ■■■+P + 1- Therefore the degree of D is p7t(.P",p)-l. The substitution E -("£•■■)• where h = i + 1, mod. p, l=h=p,i=l,2,b',---,p, has the following properties : 1) E-1A»E = Aw. Therefore the product of two substitutions of the form QE* may again be expressed in the same form. * Cf. G. A. Miller, On the Transitive Substitution Groups whose order is a power of a Prime Number, American Journal of Mathematics, vol. 23 (1901), p. 176. fCf. Netto, Theory of substitutions (translated by Cole), I 39, page 41. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 266 w. findlay: the sylow [July 2) If Q changes L^- into L¿ g, then QE* will change L^ to LA g where h m i + k, mod. p. Therefore QE*- Q'E*' if, and only if, Q = Q' and k m k, mod. p. Hence it follows that the totality of substitutions of the form QE4 constitute a group of order which is therefore a ^5P". The process of induction obtained above is to be conceived of as beginning with a = 1. The Q in this case is the identity substitution. For the full application of the induction it is required that the original pa letters be dis- tributed into p sets, which we shall call the subsets of order one, then each of these into p subsets of order two, each of the latter subsets into p subsets of order three, and so on until finally we have the subsets of order a consist- ing of one letter each. The required correspondence between the letters in the p subsets of order h + 1 into which any subset of order h is divided is to be obtained by ordering the p subdivisions at each stage. For convenience the totality of the p" letters will be spoken of as the subset of order 0. All these requirements are met by the following a-partite suffix notation : ■k{»i,»>.U» where 1 =6;=^(i = l,2, • • •, a). The first g elements in the suffix define the subset of order g to which the letter belongs, the (g + l)th suffix which of the p subsets of this division con- tains it and the remaining a — g — 1 elements serve to establish the required correspondence of these p subsets of order g + 1. Here g has the range gr=0,l,2,...,a-l. The E required to obtain a particular ^S**""'(0=^ = a — 1), (say that which acts upon the subset of order g whose letters have cx, c2, • ■-, c as the first g elements in their suffix), from the p sp,**"""""1upon the p subsets of order g + 1 contained in the above, will be "P _ "P _.( ""{»I,»».K) \ "'- <C"C2.*>"V""' LKd2.dy ■■)' where dgJrX==bgJrX+ l(moc\.p)xtbi = ci(i=l, 2,-..,g), dg+x= bg+x,if for some i, in i — 1, 2, •••, g, bf + et, df— bj (j = l,2,---,g,g + 2,---,a). Hereafter we shall use a , ß , y , etc., as symbols for gr-partite numbers with elements a., b., e\, etc. The general g'-partite number with elements having range 1, 2, 3, • • •, p will be denoted by X m {lt, lt, • • •, I ) and the totality License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1904] SUBGROUPS OF THE SYMMETRIC GROUP 267 of these, p" in number, will be denoted by A . For g = 0 we have but one E which we may denote by E0. The totality of E, 'a (g = 0, 1, •••, a — 1), in number a-l f(p,»í)-Zí?. 17=0 constitute a system of generators of a 1^°. § 4. A normal form for the general P. We shall say that X^is contained in X'hii 1) g <.h, 2) l.= l't (i = 1, 2, ■• ■,g). If X and \'h are not the same and neither one contains the other they will be said to be independent.
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